A Remark on Zhong-Yang's Eigenvalue Estimate

Abstract

Moreover we know exactly when the equality holds: if and only if D is a disc. For any optimal geometric inequality it is important to have a complete understanding of the equality case. Sometimes this can be easily achieved by checking the proof of the inequality. Take as an example the following elegant theorem due to Lichenerowicz [L]: let (M; g) be a compact Riemannian manifold of dimension n with Ric n 1, then 1 n, where 1 is the rst eigenvalue of the Laplacian operator on functions. It was proved by Obata [O] several years later that equality holds i¤ (M; g) is isometric to the standard sphere S. This is not di¢ cult to prove by tracing back each inequality in Lichnerowicz’s argument. In other cases characterizing the equality case may not be so easy. Take the Myers theorem proved in 1941: let (M; g) be a compact Riemannian manifold of dimension n with Ric n 1, then its diameter d (see e.g. [P]). To understand the equality case it is far from enough to simply analyze the proof of the inequality as doing so only gives some information along a geodesic. Some new idea is required. It was only proved in 1975 by Cheng [C] that (M; g) is isometric to the standard sphere S if d = . The proof is not easy. There is an elementary proof [Sh] using the Bishop-Gromov volume comparison theorem. For a compact Riemannian manifold (M; g) with nonnegative Ricci curvature, Li-Yau [LY, Li] proved the beautiful inequality 1 2 2d2 . By sharpening Li-Yau’s method, Zhong-Yang [ZY] improved the inequality to 1 2 d2 , which is optimal as equality holds on S. It is a natural question if S is the only case for equality. To answer this question it is not enough to go through Zhong-Yang’s proof of the inequality. This is raised as an open problem by Sakai in [S]. The main purpose of this short note is to derive the following